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I'm able to change the theme for each journal, and that works very well. However, the default theme is still active for the system running in the background. Or in other words, when clicking on 'user home', I get the greenish template again (where I use the classicRed for the actual journal). I've been looking around and it seems as if it would require a core hack, which I like to avoid as much as possible.

So any advice on how to change the theme or should I indeed modify the core files?

If so, is there a way to remove the 'User Home' without all too much effort and no core hacks?

You can't currently enable a theme at the Site level, but as a workaround you can upload a stylesheet. Just download the stylesheet of the theme you're using (it will be in plugins/themes/yourThemeName); and upload it in Site Administration -> Site Settings -> Site Style Sheet.

If you don't want people to see the Site-level pages, you can redirect to a specific journal from the same Site Settings page -- just choose the journal you want to redirect to under the Journal Redirect option. Of course, this isn't a great option for multi-journal installs.